If you're using a lens built for a full frame series and using a crop sensor its obvious that the lens crops your image for you. Some peoplle hate it I tend to like the DX sensor to full frame crop and the farther reach..but!

Does using the FX lens on my crop sensor change anything about the closest focusing distance?

As long as you remember to convert the focal length to account for the apparent change in focal length, you'll be all set. As others have mentioned, there is no actual change in focal length: the phrase usually used is something like, "A 50mm DX lens gives the same field of view as a 75mm lens on a full frame camera."

If you're using a lens built for a full frame series and using a crop sensor its obvious that the lens crops your image for you. ...

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No, the lens does nothing to "crop" the image. Relative to using the lens on a FX sensor, the DX sensor does the cropping.

Does using the FX lens on my crop sensor change anything about the closest focusing distance?

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No, nothing about the body alters ANY attribute of the lens, ever. Focal length, minimum focusing distance, f/stop, ... all remain the same. The only things that differ are things where the lens and body interact. Primarily, the angular field of view.